4 Permutation Matrices Associated to Bent Functions
95
Fig. 4.1 The permutation
matrix assigned to the
function f 2 in Example 4.7
0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0
000
0001 010
011
100
101
110
111
000
001
010
011
100
101
110
111
P =
5
It is obvious that
a 00 ⊕ b 00 ⊕ c 00 ⊕ d 00 = 000|011 ⊕ 001|000 ⊕ 100|001 ⊕ 101|010 = 000|000,
a 01 ⊕ b 01 ⊕ c 01 ⊕ d 01 = 000|100 ⊕ 001|111 ⊕ 100|110 ⊕ 101|101 = 000|000,
a 10 ⊕ b 10 ⊕ c 10 ⊕ d 10 = 010|011 ⊕ 011|000 ⊕ 110|001 ⊕ 111|010 = 000|000,
a 11 ⊕ b 11 ⊕ c 11 ⊕ d 11 = 010|100 ⊕ 011|111 ⊕ 110|110 ⊕ 111|101 = 000|000.
This permutation matrix can be specified in terms of cycles as
P = (10)(0, 9, 13, 12, 2)(1, 7, 15, 5, 6)(3, 14, 11, 4, 8).
Example 4.8 Consider the bent function f 4 in 6 variables specified by the (0, 1) →
(1, −1) encoded function vector
F 4 = [1, 1, −1, 1, −1, −1, −1, 1, −1, −1, 1, −1, −1, −1, −1, 1,
−1, −1, 1, −1, 1, 1, 1, −1, 1, 1, 1, −1, 1, 1, −1, 1,
1, 1, −1, 1, −1, −1, −1, 1, −1, −1, 1, −1, −1, −1, −1, 1,
1, 1, −1, 1, −1, −1, −1, 1, −1, −1, −1, 1, −1, −1, 1, −1]
T .
Its Gibbs derivative is
D f 4 = [30, 28, −27, 25, −20, −22, −17, 19, −26, −24, 15, −13, −16, −18, −5,
7, −62, −60, 51, −49, 52, 54, 57, −59, 56, 58, 37, −39, 50, 48, −47, 45,
14, 12, −11, 9, −4, −6, −1, 3, −10, −8, 31, −29, 0, −2, −21, 23, 46, 44,
−35, 33, −36, −38, −41, 43, −40, −42, −53, 55, −34, −32, 63, −61]
T .